Angles and the Unit Circle
Contents
Angles and the Unit Circle#
In this chapter we will define the unit circle and the various ways we make angular measurements on it.
Angle Measurements#
In cartesian space (an \(xy\)-grid) we define a ray along the \(x\) axis to be at zero degrees. We can rotate that ray counter-clockwise to an angle \(\phi\) and sweep around the \(xy\) plane.
Fig. 2 Counterclockwise rotations are by definition positive (increasing in angle).#
A complete circle is defined to be \(360^\circ\). Manipulate the slider on the next plot to get an idea of where different angular measurements fall on the circle.
Arc Length and Radians#
The circumference of a circle is
We define a radian as the angle swept out by a line equal in length to the radius of the circle.
Fig. 3 One radian is about 57.3 degrees.#
A radian is the “natural” unit for angles because it defines arcs around a circle in terms of its natural unit, \(\pi\). We can quickly convert between degrees and radians with
Given an angle \(\phi\) measured in radians, we also define arc-length \(s\) as the length on the circumference of a circle.
Arc-length as a beautifl result that helps us remember it by recognizing that we complete one complete loop, \(\phi=2\pi\), we will “travel” an arc-length equal to the circumference.
Exercises#
A car makes \(32^\circ\) turn on a road with a radius of curvature of 115 meters. How far did the car travel on that turn?
Solution
Don’t forget to convert the \(32^\circ\) to radians! Otherwise your answer will be in the wrong units.
\[s=r\phi=(115\text{m})(32^\circ)\]\[=(115\text{m})\left(32^\circ\frac{2\pi}{360^\circ}\right)\]\[= 64.23 \text{m}\]How many times will 5.5 meter long rope wrap around an axle with a diameter of 8 centimeters?
Solution
Let \(L=5.5\) m and note that this will equal some number \(N\) multiple of the circumference.
\[ L=N \cdot C \]\[ L=N \cdot 2\pi r \]Since we are given the diameter we’ll rewrite the previous equation using diameter, then solve for the number of turns \(N\).
\[ L=N\cdot \pi D \]\[ N=\frac{L}{\pi D}\]\[ N=\frac{(5.5\,\text{m})}{\pi (0.08\,\text{m})} \]\[ N=21.8 \]